Diffusion–annihilation dynamics in one spatial dimension

Constructing Integrable Lindblad Superoperators

We develop a new method for the construction of one-dimensional integrable Lindblad systems, which describe quantum many body models in contact with a Markovian environment. We find several new models with interesting features, such as annihilation-diffusion processes, a mixture of coherent and classical particle propagation, and a rectified steady state current. We also find new ways to represent known classical integrable stochastic equations by integrable Lindblad operators. Our method can be extended to various other situations and it establishes a structured approach to the study of solvable open quantum systems.

Effect of bias in a reaction-diffusion system in two dimensions

We consider a single species reaction diffusion system on a two-dimensional lattice where the particles A are biased to move towards their nearest neighbors and annihilate as they meet. Allowing the bias to take both negative and positive values parametrically, any nonzero bias is seen to drastically affect the behavior of the system compared to the unbiased (simple diffusive) case. For positive bias, a finite number of dimers, which are isolated pairs of particles occurring as nearest neighbors, exist while for negative bias, a finite density of particles survives. Both the quantities vanish in a power-law manner close to the diffusive limit with different exponents. The appearance of dimers is exclusively due to the parallel updating scheme used in the simulation. The results indicate the presence of a continuous phase transition at the diffusive point. In addition, a discontinuity is observed at the fully positive bias limit. The persistence behavior is also analysed for the system.

Integrable dissipative exclusion process: Correlation functions and physical properties

We study a one-parameter generalization of the symmetric simple exclusion process on a one-dimensional lattice. In addition to the usual dynamics (where particles can hop with equal rates to the left or to the right with an exclusion constraint), annihilation and creation of pairs can occur. The system is driven out of equilibrium by two reservoirs at the boundaries. In this setting the model is still integrable: it is related to the open XXZ spin chain through a gauge transformation. This allows us to compute the full spectrum of the Markov matrix using Bethe equations. We also show that the stationary state can be expressed in a matrix product form permitting to compute the multipoints correlation functions as well as the mean value of the lattice and the creation-annihilation currents. Finally, the variance of the lattice current is computed for a finite-size system. In the thermodynamic limit, it matches the value obtained from the associated macroscopic fluctuation theory.

A+A→∅ model with a bias towards nearest neighbor

We have studied the A+A→∅ reaction-diffusion model on a ring, with a bias ε(0≤ε≤0.5) of the random walkers A to hop towards their nearest neighbor. Though the bias is local in space and time, we show that it alters the universality class of the problem. The z exponent, which describes the growth of average spacings between the walkers with time, changes from the value 2 at ε=0 to the mean-field value of unity for any nonzero ε. We study the problem analytically using independent interval approximation and compare the scaling results with those obtained from simulation. The distribution P(k,t) (per site) of the spacing between two walkers is given by t(-2/z)f(k/t(1/z)) and is obtained both analytically and numerically. We also obtain the result that εt becomes the new time scale for ε≠0.

Scaling approach to related disordered stochastic and free-fermion models

Motivated by mapping from a stochastic system with spatially random rates, we consider disordered non-conserving free-fermion systems using a scaling procedure for the equations of motion. This approach demonstrates disorder-induced localization acting in competition with the asymmetric driving. We discuss the resulting implications for the original stochastic system.

Exact multipoint and multitime correlation functions of a one-dimensional model of adsorption and evaporation of dimers

In this work, we provide a method that allows us to compute exactly the multipoint and multitime correlation functions of a one-dimensional stochastic model of dimer adsorption evaporation with random (uncorrelated) initial states. In particular, explicit expressions of the two-point non-instantaneous/instantaneous correlation functions are obtained. The long-time behavior of these expressions is discussed in detail and in various physical regimes.

Two-point correlation functions of the diffusion-limited annihilation in one dimension

Two-point density-density correlation functions for the diffusive binary reaction system A+A--> phi are obtained in one dimension via Monte Carlo simulation. The long-time behavior of these correlation functions clearly deviates from that of a recent analytical prediction of Bares and Mobilia [Phys. Rev. Lett. 83, 5214 (1999)]. An alternative expression for the asymptotic behavior is conjectured from numerical data.

Exact solution of a class of one-dimensional nonequilibrium stochastic models

We consider various one-dimensional nonequilibrium models, namely, the diffusion-limited pair-annihilation and creation model (DPAC) and its unbiased version (the Lushnikov model), the DPAC model with particle injection, as well as (biased) diffusion-limited coagulation model (DC). We study the DPAC model using an approach based on a duality transformation and the generating function of the dual model. We are able to compute exactly the density and correlation functions in the general case with arbitrary initial states. Further, we assume that a source injects particles in the system. Solving, via the duality transformation, the equations of motion of the density, and the noninstantaneous two-point correlation functions, we see how the source affects the dynamics. Finally we extend the previous results to the DC model with help of a similarity transformation.

Exact density profile of a stochastic reaction-diffusion process

We calculate exactly the time dependent density profile of a one-dimensional stochastic reaction-diffusion process of hard-core particles subjected to the reactions AA <--> OO and AO <--> OA. The solution is based on the fundamental property that the evolution operator, defined over an appropriate vector space, transforms vectors with n kinks into vectors with n or n+2 kinks, only. In this space, a basis vector is represented by a string of plus and minus signs and a kink is defined as a pair of opposite signs. The exact time dependent profiles are calculated for the cases of uncorrelated initial states that are translational invariant as well as initial states that are inhomogeneous in space.